Existence and stability of traveling waves for discrete nonlinear Schroedinger equations over long times

TL;DR

This paper investigates the existence and long-term stability of solitary traveling wave solutions in the one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity, especially near the continuous limit.

Contribution

It constructs solutions close to continuous traveling waves and proves their stability over long times, extending understanding of DNLS dynamics.

Findings

Existence of discrete traveling wave solutions near continuous waves

Long-time stability of these solutions

Description of dynamics near solutions using modulation methods

Abstract

We consider the problem of existence and stability of solitary traveling waves for the one dimensional discrete non linear Schroedinger equation (DNLS) with cubic nonlinearity, near the continuous limit.We construct a family of solutions close to the continuous traveling waves and prove their stability over long times. Applying a modulation method, we also show that we can describe the dynamics near these discrete traveling waves over long times.